Let A be the free abelian group on a set X. If X is not a singleton then A is not free on X in the category of groups.
How can I show that A is not free on some other set Y in the category of groups?
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Let $A$ be an abelian group containing $Y$; suppose $x$ and $y$ are distinct elements of $Y$ and consider your favorite non abelian group $G$, with $g,h\in G$ so that $gh\ne hg$.
Does there exist a homomorphism $A\to G$ such that $x\mapsto g$ and $y\mapsto h$?
So can an abelian group be free on a set with more than one element?
A free group with one generator is cyclic, which $A$ is not. A free group with more than one generator is non-abelian, while $A$ is abelian. Thus $A$ can be neither of those things.